Neural Network Braking Control

1. System Modeling

The ABS system was modeled using the quarter-car model describing vehicle, wheel, and braking system dynamics. The slip ratio $\lambda$, mechanical and aerodynamic quantities are related by:

$$\lambda = \frac{v - r\omega}{v}$$

Figure 1 - Simplified Quarter-Car Model

The following differential equations describe the system:

• Vehicle dynamics:

  $$M\dot{v} = -\mu(\lambda)F_z - C_x v^2$$

• Wheel dynamics:

  $$I\dot{\omega} = \mu(\lambda)F_z r - B\omega - \tau_b$$

• Braking system:

  $$\dot{\tau}_b = \frac{-\tau_b + K_b u}{\tau}$$

The nonlinear relationship between slip ratio and friction coefficient $\mu(\lambda)$ is modeled by a law of type:

$$\mu(\lambda) = \frac{2\mu_0 \lambda_0 |\lambda|}{\lambda_0^2 + \lambda^2}$$

with maximum grip peak around $\lambda_0 = 0.25$.

2. NARMA-L2 Controller

The NARMA-L2 controller linearizes a nonlinear system using two neural networks (called f-network and g-network), trained from past outputs and control inputs:

$$y(k+d) = f(\cdot) + g(\cdot) \cdot u(k+1)$$

The control law is derived as:

$$u(k+1) = \frac{y_{\text{ref}} - f_k}{g_k}$$

Where $y_{ref} = 0.25$ represents the optimal slip ratio.

The networks used have a simple single hidden layer architecture, with $tanh$ function, and take as input:

$$x(k)=[y(k),y(k−1),y(k−2),u(k),u(k−1),u(k−2)]$$

3. Data Generation and Training

To train the networks, a set of 50 simulated braking scenarios was generated using a PID controller. Parameters such as initial speed, braking torque, or vehicle state were randomly chosen to vary conditions.

Parameter	Range
Initial speed	20–40 m/s
Initial slip	0.1–0.4
Initial braking torque	0–20 Nm

Figure 2 - Generated Data Analysis

The networks were trained on approximately 20,000 data points with a standard method (Adam, 5 to 10 epochs). The objective was to accurately predict the system response and extract a stable control law.

Figure 3 - Learning Loss for Both Neural Networks

4. Experimental Results

The NARMA-L2 controller was compared to the classical PID controller:

• Slip ratio tracking: PID remains more accurate and stable. NARMA-L2 shows significant deviations around the setpoint.
• Braking effort: NARMA-L2 applies lower, sometimes unstable braking pressures, but sufficient for comparable deceleration.
• Mechanical stress reduction: Gentler braking efforts can extend component lifespan.

Figure 4 - Performance Comparison: Slip and Braking Torque

5. Discussion and Perspectives

Even though the NARMA-L2 controller did not outperform PID in terms of accuracy, it presents interest for:

• brake wear reduction,
• energy savings in electromechanical systems,
• applications in electric and autonomous vehicles.

The project also highlighted the importance of training data quality: exclusive use of a PID as data generator can bias learning.